16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 9.3.5

Operators on Inner Product Spaces

Mathematics — Linear Algebra · positive-matrices

Problem

Let A=[1121234].A = \begin{bmatrix} 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{3}{4} \end{bmatrix}. (a) Show that AA is positive. (b) Let VV be the space of 2×12 \times 1 real matrices, with the inner product (X∣Y)=YtAX(X|Y) = Y^t A X. Find an orthonormal basis for VV, by applying the Gram-Schmidt process to the basis {X1,X2}\{X_1, X_2\} defined by X1=[10],X2=[01].X_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad X_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. (c) Find an invertible 2×22 \times 2 real matrix PP such that A=PtPA = P^t P.

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